 1.4.1E: Compute the products in Exercises 1–4 using (a) the definition, as ...
 1.4.2E: Compute the products using (a) the definition, as in Example , and ...
 1.4.3E: Compute the products using (a) the definition, as in Example , and ...
 1.4.4E: Compute the products using (a) the definition, as in Example , and ...
 1.4.5E: Use the definition of Ax to write the matrix equation as a vector e...
 1.4.6E: Use the definition of Ax to write the matrix equation as a vector e...
 1.4.7E: In Exercises 5–8, use the definition of Ax to write the matrix equa...
 1.4.8E: Use the definition of Ax to write the matrix equation as a vector e...
 1.4.9E: write the system first as a vector equation and then as a matrix eq...
 1.4.10E: write the system first as a vector equation and then as a matrix eq...
 1.4.11E: Given A and b write the augmented matrix for the linear system that...
 1.4.12E: Given A and b write the augmented matrix for the linear system that...
 1.4.13E: Is u in the plane in ?3 spanned by the columns of A? (See the figur...
 1.4.14E: Let . Is u in the subset of R3 spanned by the columns of A? Why or ...
 1.4.15E: Let . Show that the equation Ax = b does not have a solution for al...
 1.4.16E: Repeat ExerciseExercise Show that the equation Ax = b does not have...
 1.4.17E: Exercises 17–20 refer to the matrices A and B below. Make appropria...
 1.4.18E: Exercises 17–20 refer to the matrices A and B below. Make appropria...
 1.4.19E: Exercises 17–20 refer to the matrices A and B below. Make appropria...
 1.4.20E: Exercises 17–20 refer to the matrices A and B below. Make appropria...
 1.4.21E: Does {v1, v2, v3} span ?4 ? Why or why not?
 1.4.22E: Let Does {v1, v2, v3} span ?3? Why or why not?
 1.4.23E: a. The equation Ax = b is referred to as a vector equation.b. A vec...
 1.4.24E: a. Every matrix equation Ax = b corresponds to a vector equation wi...
 1.4.25E: Note that Use this fact (and no row operations) to find scalars c1,...
 1.4.26E: Let It can be shown that 3u  5v  w = 0. Use this fact (and no row...
 1.4.27E: Let q1, q2, q3, and v represent vectors in ?5, and let x1, x2, and ...
 1.4.28E: Rewrite the (numerical) matrix equation below in symbolic form as a...
 1.4.29E: Construct a 3 × 3 matrix, not in echelon form, whose columns span ?...
 1.4.30E: Construct a 3 × 3 matrix, not in echelon form, whose columns do not...
 1.4.31E: Let A be a 3 × 2 matrix. Explain why the equation Ax = b cannot be ...
 1.4.32E: Could a set of three vectors in R4 span all of ?4? Explain. What ab...
 1.4.33E: Suppose A is a 4 × 3 matrix and b is a vector in ?4 with the proper...
 1.4.34E: Suppose A is a 4 × 3 matrix and b is a vector in ?4 with the proper...
 1.4.35E: Let A be a 3 × 4 matrix, let yl and y2 be vectors in ?3, and let w ...
 1.4.36E: Let A be a 5 × 3 matrix, let y be a vector in ?3, and let z be a ve...
 1.4.37E: [M] In Exercises 37–40, determine if the columns of the matrix span...
 1.4.38E: Determine if the columns of the matrix span ?4.
 1.4.39E: Determine if the columns of the matrix span ?4.
 1.4.40E: Determine if the columns of the matrix span ?4.
 1.4.41E: [M] Find a column of the matrix in Exercise 39 that can be deleted ...
 1.4.42E: [M] Find a column of the matrix in Exercise 40 that can be deleted ...
Solutions for Chapter 1.4: Linear Algebra and Its Applications 5th Edition
Full solutions for Linear Algebra and Its Applications  5th Edition
ISBN: 9780321982384
Solutions for Chapter 1.4
Get Full SolutionsChapter 1.4 includes 42 full stepbystep solutions. Since 42 problems in chapter 1.4 have been answered, more than 40831 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Linear Algebra and Its Applications was written by and is associated to the ISBN: 9780321982384. This textbook survival guide was created for the textbook: Linear Algebra and Its Applications , edition: 5.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.